International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 188-189   | 1 | 2 |

Section 1.5.5.3. Figures and tables

M. I. Aroyoa* and H. Wondratschekb

aDepartamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

1.5.5.3. Figures and tables

| top | pdf |

Arithmetic crystal classes [m\overline{3}mI] and [m\overline{3}I]: The reciprocal lattice of a cubic lattice [I] is a cubic lattice F. Its Brillouin zone is a rhombic dodecahedron and has 12 faces, 24 edges and 14 apices, the coordinates of which are the six permutations of [\pm{{1}\over{2}},0,0] and the eight coordinate triplets of [\pm{{1}\over{4}},\pm{{1}\over{4}},\pm{{1}\over{4}}]. Eleven of these 14 points are visible in the applied projection.

The figure for arithmetic crystal class [m\overline{3}mI] is shown in Fig. 1.5.5.1[link] and the corresponding table is Table 1.5.5.1[link]. The figure for arithmetic crystal class [m\overline{3}I] is shown in Fig. 1.5.5.2[link] and the corresponding table is Table 1.5.5.2[link].

Table 1.5.5.1| top | pdf |
List of k-vector types for arithmetic crystal class [m\overline{3}mI]

See Fig. 1.5.5.1[link]. Parameter relations: [x=\textstyle{1 \over 2}\beta + \textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0]   [4 \quad a \quad m\overline3m] 0, 0, 0
[H \quad \textstyle{1 \over 2},-\textstyle{1 \over 2},\textstyle{1 \over 2}]   [4 \quad b \quad m\overline3m] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4}]   [8 \quad c \quad\overline43m ] [\textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4} ]
[N\quad 0,0,\textstyle{1 \over 2}]   [24 \quad d \quad m.mm] [\textstyle{1 \over 4},\textstyle{1 \over 4},0 ]
[\Delta \quad \alpha,-\alpha,\alpha]   [24 \quad e \quad 4m.m] [0,y,0]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}]
[\Lambda \quad \alpha,\alpha,\alpha] ex [32 \quad f \quad .3m] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F \quad \textstyle{1 \over 2}-\alpha,-\textstyle{1 \over 2}+ 3\alpha,\textstyle{1 \over 2}-\alpha] ex [32 \quad f \quad .3m] [x,\textstyle{1 \over 2}-x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F\sim F_1=[P\,R]]     [x,x,x]: [\textstyle{1 \over 4} \,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[F\sim F_3=[P\,H_2]]     [x,x,\textstyle{1 \over 2}-x]: [0 \,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[\Lambda\cup F_1=[\Gamma R]\backslash [P] ]   [32 \quad f \quad .3m] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}, x\neq\textstyle{1 \over 4}]
[D\quad \alpha,\alpha,\textstyle{1 \over 2}-\alpha]   [48 \quad g \quad 2.mm] [\textstyle{1 \over 4},\textstyle{1 \over 4},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma \quad 0,0,\alpha]   [48 \quad h \quad m.m2] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[G\quad \textstyle{1 \over 2}-\alpha,-\textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}]   [48 \quad i \quad m.m2] [x,\textstyle{1 \over 2}-x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A \quad \alpha, -\alpha, \beta]   [96 \quad j \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[B\quad \alpha+\beta,-\alpha+\beta,\textstyle{1 \over 2}-\beta] ex [96 \quad k \quad ..m] [x,\textstyle{1 \over 2}-x,z]: [0\,\lt \, z\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[B\sim B_1=[P\,N_2\,H_2]]     [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}-x\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[C \quad \alpha,\alpha,\beta] ex [96 \quad k \quad ..m] [x,x,z]: [0\,\lt \, z\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[J \quad \alpha, \beta, \alpha] ex [96\quad k \quad ..m] [x,y,x]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[J\sim J_1=[\Gamma\,P\,H_2]]     [x,x,z]: [0\,\lt \, x\,\lt \, z\,\lt \, \textstyle{1 \over 2}-x]
[C\cup B_1\cup J_1=[\Gamma\,N\,N_2\,H_2]\backslash[\Lambda,\,F_3]]   [96 \quad k \quad ..m] [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}, 0\,\lt \, z\,\lt \, \textstyle{1 \over 2}] with [z\neq x, z\neq\textstyle{1 \over 2}-x]
[GP \quad \alpha,\beta,\gamma]   [192 \quad l \quad 1] [x,y,z]: [0\,\lt \, z\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]

Table 1.5.5.2| top | pdf |
List of k-vector types for arithmetic crystal class [m\overline{3}I]

See Fig. 1.5.5.2[link]. Parameter relations: [x=\textstyle{1 \over 2}\beta + \textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma\quad 0,0,0]   [4 \quad a \quad m\overline3.] [0,0,0]
[H \quad \textstyle{1 \over 2},-\textstyle{1 \over 2},\textstyle{1 \over 2}]   [4 \quad b \quad m\overline3.] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]   [8 \quad c \quad 23.] [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]
[N \quad 0,0,\textstyle{1 \over 2}]   [24 \quad d \quad 2/m..] [\textstyle{1 \over 4},\textstyle{1 \over 4},0]
[\Delta\quad \alpha,-\alpha,\alpha]   [24 \quad e \quad mm2..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[\Lambda \quad \alpha,\alpha,\alpha] ex [32 \quad f \quad .3.] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F \quad \textstyle{1 \over 2}-\alpha, -\textstyle{1 \over 2}+3\alpha, \textstyle{1 \over 2}-\alpha] ex [32 \quad f \quad .3.] [x,\textstyle{1 \over 2}-x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[F\sim F_1 =[P\,R]]     [x,x,x]: [\textstyle{1 \over 4}\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Lambda \cup F_1\sim[\Gamma\, R]\backslash[P]]   [32 \quad f\quad .3.] [x,x,x]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}, x\neq\textstyle{1 \over 4}]
[D\quad \alpha,\alpha,\textstyle{1 \over 2}-\alpha]   [48 \quad g \quad 2..] [\textstyle{1 \over 4},\textstyle{1 \over 4},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma\quad 0,0,\alpha] ex [48 \quad h \quad m..] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[G \quad\textstyle{1 \over 2}-\alpha,-\textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}] ex [48 \quad h \quad m..] [x,\textstyle{1 \over 2}-x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A = [\Gamma\, N\, H] \quad \alpha,-\alpha,\beta] ex [48\quad h \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[AA = [\Gamma\,H_2\,N] \quad {-\alpha},\alpha,\beta] ex [48 \quad h \quad m..] [x,y,0]: [0\,\lt \, y\,\lt \, x\,\lt \, \textstyle{1 \over 2}-y]
[\Sigma\cup G\cup A\cup AA]   [48 \quad h \quad m..] [x,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x\,\lt \, \textstyle{1 \over 2}\ \cup] [\cup \ x,\textstyle{1 \over 2}-x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4} ]
[GP \quad \alpha,\beta,\gamma]   [96 \quad i \quad 1] [x,y,z]: [0\,\lt \, z\leq x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x \ \cup] [\cup \ x,y,z]: [0\,\lt \, z\,\lt \, y\,\lt \, x\leq\textstyle{1 \over 2}-y\ \cup] [\cup \ x,x,z]: [0\,\lt \, z\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[Figure 1.5.5.1]

Figure 1.5.5.1 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [m\overline{3}mI]. Space groups: [Im\overline{3}m-O_h^9] (229), [Ia\overline{3}d-O_h^{10}] (230). Reciprocal-space group ([Fm\overline{3}m])*, No. 225 (see Table 1.5.5.1[link]). The representation domain of CDML is identical with the asymmetric unit. Auxiliary points: R: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2}]; N2: [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 2}]; H2: [0,0,\textstyle{1 \over 2}]. Flagpole: [F_1=[P\,R]\quad x,x,x]: [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}]. Wing: [B_1 \cup J_1=[\Gamma\,P\,N_2\,H_2]\backslash[F_3]] [ x,x,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]; [x \,\lt \, z \,\lt \,\textstyle{1 \over 2}] with [z \neq \textstyle{1 \over 2} - x].

[Figure 1.5.5.2]

Figure 1.5.5.2 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [m\overline{3}I]. Space groups [Im\overline{3}-T_h^5] (204), [Ia\overline{3}-T_h^7] (206). Reciprocal-space group ([Fm\overline{3}])*, No. 202 (see Table 1.5.5.2[link]). The representation domain of CDML is identical with the asymmetric unit. Auxiliary points: R: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2}]; H2: [\textstyle{1 \over 2},0,0]. Flagpole: [F_1=[P\,R]] [ x,x,x]: [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}].

Arithmetic crystal class [4/mmmI]: There are two different types of Brillouin zones for the tetragonal I lattice, one for [c \,\lt\, a] (Fig. 1.5.5.3[link], Table 1.5.5.3[link]) and one for [c\,\gt\, a] (Fig. 1.5.5.4[link], Table 1.5.5.4[link]). The first type of Brillouin zone, Fig. 1.5.5.3[link], is a tetragonal elongated rhombdodecahedron with 12 faces, four of them being hexagons. There are 18 apices; 14 of them are visible. The Brillouin zone of Fig. 1.5.5.4[link] is a tetragonally deformed cuboctahedron with 14 faces. There are 24 apices; 18 of them are visible.

Table 1.5.5.3| top | pdf |
List of k-vector types for arithmetic crystal class [4/mmmI]: [c/a \,\lt\, 1]

See Fig. 1.5.5.3[link]. Wyckoff positions e and f exchanged. Parameter relations: [x=-\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta+\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0]   [2 \quad a \quad 4/mmm] [0,0,0]
[M \quad {-\textstyle{1 \over 2}},\textstyle{1 \over 2},\textstyle{1 \over 2}]   [2 \quad b \quad 4/mmm] [\textstyle{1 \over 2},\textstyle{1 \over 2},0]
[M\sim M_2]     [0,0,\textstyle{1 \over 2}]
[X \quad 0,0,\textstyle{1 \over 2}]   [4 \quad c \quad mmm.] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4}]   [4 \quad d \quad \overline 4m2] [0,\textstyle{1 \over 2},\textstyle{1 \over 4}]
[N \quad 0, \textstyle{1 \over 2}, 0]   [8 \quad f\quad ..2/m] [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]
[\Lambda\quad \alpha, \alpha, -\alpha] ex [4 \quad e \quad 4mm] [0,0,z]: [0\,\lt \, z\leq z_0]
[V \quad {-\textstyle{1 \over 2}} + \alpha,\textstyle{1 \over 2} + \alpha,\textstyle{1 \over 2} -\alpha] ex [4 \quad e \quad 4mm] [\textstyle{1 \over 2},\textstyle{1 \over 2},z]: [0\,\lt \, z\,\lt \, z_2=\textstyle{1 \over 2}-z_0]
[V \sim \Lambda_1=[Z_0\,M_2]]     [0,0,z]: [\,z_0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[\Lambda \cup \Lambda_1 = [\Gamma\,M_2]]   [4 \quad e \quad 4mm] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[W \quad \alpha, \alpha, \textstyle{1 \over 2}-\alpha]   [8\quad g \quad 2mm.] [0,\textstyle{1 \over 2},z]: [\, 0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma \quad{-\alpha}, \alpha, \alpha]   [8 \quad h \quad m.2m] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Delta \quad 0,0,\alpha]   [8 \quad i \quad m2m.] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[Y \quad {-\alpha}, \alpha, \textstyle{1 \over 2}]   [8 \quad j \quad m2m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[Q \quad\textstyle{1 \over 4}-\alpha,\textstyle{1 \over 4}+\alpha,\textstyle{1 \over 4}-\alpha]   [16 \quad k \quad ..2] [x,\textstyle{1 \over 2}-x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[C \quad {-\alpha},\alpha,\beta]   [16 \quad l \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B \quad \alpha,\beta,-\alpha]   [16 \quad m \quad ..m] [x,x,z]: [[\Gamma\,M\,Z_2\,Z_0]]
[B=B_1\cup B_2] = [[\Gamma\,M\,Z_2\,N\,T]\cup [T\,N\,Z_0]]      
[B_2 \sim B_3]     [x,x,z]: [[N\,Z_2\,T_2]]
[B_1\cup B_3=[\Gamma\,M\,T_2\,T]]   [16 \quad m \quad ..m] [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2},\,0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A \quad \alpha,\alpha,\beta] ex [16 \quad n \quad .m.] [0,y,z]: [[\Gamma\,X\,P\,Z_0]]
[E\quad \alpha - \beta, \alpha+\beta, \textstyle{1 \over 2}-\alpha] ex [16 \quad n \quad .m.] [x,\textstyle{1 \over 2},z]: [[M\,X\,P\,Z_2]]
[E \sim A_1]     [0,y,z]: [[P\,X_2\,M_2\,Z_0]]
[A \cup A_1=[\Gamma\,X\,X_2\,M_2]]   [16 \quad n \quad .m.] [0,y,z]: [0\,\lt \, y,z\,\lt \, \textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [32 \quad o \quad 1] [x,y,z]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,y,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]

Table 1.5.5.4| top | pdf |
List of k-vector types for arithmetic crystal class [4/mmmI]: [c/a \,\gt\, 1]

See Fig. 1.5.5.4[link]. Wyckoff positions [e] and [f] exchanged. Parameter relations: [x=-\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta], [y=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta+\gamma], [z=\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0]   [2 \quad a \quad 4/mmm] [0,0,0]
[M \quad \textstyle{1 \over 2}, \textstyle{1 \over 2}, -\textstyle{1 \over 2}]   [2 \quad b \quad 4/mmm] [0,0,\textstyle{1 \over 2}]
[M\sim M_2]     [\textstyle{1 \over 2},\textstyle{1 \over 2},0]
[X \quad 0,0,\textstyle{1 \over 2}]   [4 \quad c \quad mmm.] [0,\textstyle{1 \over 2},0]
[P \quad \textstyle{1 \over 4}, \textstyle{1 \over 4}, \textstyle{1 \over 4}]   [4 \quad d \quad \overline 4m2] [0,\textstyle{1 \over 2},\textstyle{1 \over 4}]
[N \quad 0, \textstyle{1 \over 2}, 0]   [8 \quad f \quad ..2/m] [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]
[\Lambda\quad \alpha, \alpha, -\alpha]   [4 \quad e \quad 4mm] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[W \quad \alpha, \alpha, \textstyle{1 \over 2}-\alpha]   [8 \quad g \quad 2mm.] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}]
[\Sigma\quad {-\alpha}, \alpha, \alpha] ex [8 \quad h \quad m.2m] [x,x,0]: [0\,\lt \, x\leq s_2]
[F \quad \textstyle{1 \over 2}-\alpha,\textstyle{1 \over 2}+\alpha,-\textstyle{1 \over 2}+\alpha] ex [8\quad h \quad m.2m] [x,x,\textstyle{1 \over 2}]: [0\,\lt \, x\,\lt \, s=\textstyle{1 \over 2}-s_2]
[F\sim\Sigma_1=[S_2\,M_2]]     [x,x,0]: [s_2\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Sigma\cup \Sigma_1=[\Gamma M_2]]   [8 \quad h \quad m.2m] [x,x,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[\Delta\quad 0,0,\alpha]   [8 \quad i \quad m2m.] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[Y\quad {-\alpha}, \alpha, \textstyle{1 \over 2}] ex [8 \quad j \quad m2m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\leq r]
[U \quad \textstyle{1 \over 2},\textstyle{1 \over 2},-\textstyle{1 \over 2}+\alpha] ex [8 \quad j \quad m2m.] [0,y, \textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, g=\textstyle{1 \over 2}-r]
[U\sim Y_1=[R\,M_2]]     [x,\textstyle{1 \over 2},0]: [r\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[Y\cup Y_1=[X\,M_2]]   [8 \quad j \quad m2m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[Q\quad\textstyle{1 \over 4}-\alpha,\textstyle{1 \over 4}+\alpha,\textstyle{1 \over 4}-\alpha]   [16 \quad k \quad ..2] [x,\textstyle{1 \over 2}-x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[C\quad {-\alpha},\alpha,\beta] ex [16 \quad l \quad m..] [x,y,0]: [[\Gamma\,S_2\,R\,X]]
[D \quad\textstyle{1 \over 2}-\alpha,\textstyle{1 \over 2}+\alpha,-\textstyle{1 \over 2}+\beta] ex [16 \quad l \quad m..] [x,y,\textstyle{1 \over 2}]: [[M\,S\,G]]
[D\sim C_1]     [x,y,0]: [[M_2\,R\,S_2]]
[C\cup C_1=[\Gamma\,M_2\,X]]   [16 \quad l \quad m..] [x,y,0]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B\quad\alpha,\beta,-\alpha]   [16 \quad m \quad ..m] [x,x,z]: [[\Gamma\,S_2\,S\,M]]
[B=B_1\cup B_2] = [[\Gamma\,S_2\,N\,T]\cup[T\,N\,S\,M]]      
[B_2\sim B_3]     [x,x,z]: [[T_2\,N\,S_2\,M_2]]
[B_1\cup B_3=[\Gamma\,M_2\,T_2\,T]]   [16 \quad m \quad ..m] [x,x,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2},0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,x,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 4}]
[A\quad\alpha,\alpha,\beta] ex [16 \quad n \quad .m.] [0,y,z]: [[\Gamma\, X\,P\,G\,M]]
[E \quad\alpha - \beta, \alpha+\beta, \textstyle{1 \over 2}-\alpha] ex [16 \quad n \quad .m.] [x,\textstyle{1 \over 2},z]: [[X\,P\,R]]
[E\sim A_1]     [0,y,z]: [[X_2\,G\,P]]
[A\cup A_1=[\Gamma\,X\,X_2\,M]]   [16 \quad n \quad .m.] [0,y,z]: [0\,\lt \, y,z\,\lt \, \textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [32 \quad o \quad 1] [x,y,z]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z\,\lt \, \textstyle{1 \over 4}\ \cup] [\cup \ x,y,\textstyle{1 \over 4}]: [0\,\lt \, x\,\lt \, y\,\lt \, \textstyle{1 \over 2}-x]
[Figure 1.5.5.3]

Figure 1.5.5.3 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [4/mmmI]: [c/a \,\lt \, 1]. Space groups [I4/mmm-D_{4h}^{17}] (139) to [I4_1/acd-D_{4h}^{20}] (142). Reciprocal-space group ([I4/mmm])*, No. 139: [c^{*}/a^{*}\,\gt \, 1] (see Table 1.5.5.3[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T: [0,0,\textstyle{1 \over 4}]; T2: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 4}]; X2: [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]. Flagpole: [[T\,M_2]\quad 0,0,z]: [\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}]. Wing: [[T\,P\,X_2\,M_2]\quad 0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}], [\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}].

[Figure 1.5.5.4]

Figure 1.5.5.4 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [4/mmmI]: [c/a\,\gt \, 1]. Space groups [I4/mmm-D_{4h}^{17}] (139) to [I4_1/acd-D_{4h}^{20}] (142). Reciprocal-space group ([I4/mmm])*, No. 139: [c^{*}/a^{*} \,\lt \, 1] (see Table 1.5.5.4[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: X2: [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]; T: [0,0,\textstyle{1 \over 4}]; T2: [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 4}]. Flagpole: [[T\,M]\quad 0,0,z]: [\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}]. Wing: [[T\,P\,X_2\,M]\quad 0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2},\textstyle{1 \over 4} \,\lt \, z \,\lt \, \textstyle{1 \over 2}].

Arithmetic crystal class [mm2F]: Depending on the lattice ratios a:b:c, there are four figures in CDML for the Brillouin zone of an orthorhombic crystal with an F lattice, see Fig. 3.6 on p. 26 in CDML. Only three of them are really necessary. Therefore, the case [b^{-2}\,\gt \, c^{-2}+a^{-2}] of Fig. 3.6(c) of CDML has been omitted in these examples; it is obtained from [a^{-2}\,\gt \, c^{-2}+b^{-2}] of Figure 3.6(d) by a rotation by 90° about the c* axis. The three remaining Brillouin zones are displayed in Fig. 1.5.5.5[link] (see also Table 1.5.5.5[link]), Fig. 1.5.5.6[link] (see also Table 1.5.5.6[link]) and Fig. 1.5.5.7[link] (see also Table 1.5.5.7[link]). Fig. 1.5.5.5[link] is a distorted cuboctahedron with 14 faces, 36 edges and 24 apices, 18 of which are visible. The Brillouin zones of Figs. 1.5.5.6[link] and 1.5.5.7[link] are distorted elongated rhombdodecahedra. There are 12 faces, 28 edges and 18 apices; 14 of them are visible.

Table 1.5.5.5| top | pdf |
List of k-vector types for arithmetic crystal class [mm2F]: [a^{-2} \,\lt\, b^{-2}+c^{-2}], [b^{-2} \,\lt\, c^{-2}+a^{-2}] and [c^{-2} \,\lt\, a^{-2}+b^{-2}]

See Fig. 1.5.5.5[link]. Parameter relations: [x=-\textstyle{1 \over 2}\alpha+\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2} \alpha-\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta-\textstyle{1 \over 2}\gamma].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0] ex [2 \quad a \quad mm2] [0,0,0]
[Z \quad \textstyle{1 \over 2},\textstyle{1 \over 2} ,0] ex [2 \quad a \quad mm2] [0,0,\textstyle{1 \over 2}]
[\Lambda \quad \alpha, \alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[LE \quad {-\alpha}, -\alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[\Gamma \cup \Lambda \cup Z \cup LE]   [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[T\quad 0,\textstyle{1 \over 2},\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,0]
[T \sim T_2]     [0,\textstyle{1 \over 2} ,\textstyle{1 \over 2}]
[Y \quad \textstyle{1 \over 2}, 0, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},0]
[G \quad \alpha, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,z]: [0\,\lt \, z\leq g_0]
[G \sim H_3=[H_2\,T_4]]     [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \leq -\textstyle{1 \over 2}+g_0=h_2]
[GA \quad {-\alpha}, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}] ex [2 \quad b\quad mm2] [\textstyle{1 \over 2},0,z]: [g_2=-g_0\,\lt \, z\,\lt \, 0]
[GA \sim H_1=[H_0\,T_2]]     [0,\textstyle{1 \over 2},z]: [\textstyle{1 \over 2}-g_0=h_0 \,\lt \, z\,\lt \,\textstyle{1 \over 2}]
[H \quad \textstyle{1 \over 2}+\alpha, \alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z \leq h_0]
[HA \quad \textstyle{1 \over 2}-\alpha,-\alpha,\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [h_2=-h_0\,\lt \, z\,\lt \, 0]
[T_2 \cup H_1 \cup H \cup Y \cup HA \cup H_3]   [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[\Sigma \quad 0,\alpha,\alpha] ex [4 \quad c \quad .m.] [x,0,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[A \quad \textstyle{1 \over 2},\textstyle{1 \over 2}+\alpha,\alpha] ex [4 \quad c \quad .m.] [x,0,\textstyle{1 \over 2}]: [0\,\lt \, x\leq a_0]
[C \quad \textstyle{1 \over 2},\alpha,\textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, c_0=\textstyle{1 \over 2} -a_0]
[C \sim A_1]     [x,0,\textstyle{1 \over 2}]: [\textstyle{1 \over 2}-a_0=c_0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[J \quad \alpha, \alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\, Z\,A_0\,G_0\,T]]
[JA \quad {-\alpha}, -\alpha+\beta, \beta] ex [4\quad c \quad .m.] [x,0,z]: [[\Gamma\,T\,G_2\,A_2\,Z_2]]
[K \quad \textstyle{1 \over 2}+\alpha, \alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,H_0\,C_0]]
[K \sim J_1]     [x,0,z]: [[Y_4\,G_2\,A_2]]
[KA \quad \textstyle{1 \over 2}-\alpha, -\alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,C_0\,H_2]]
[KA \sim J_3]     [x,0,z]: [[Y_2\,G_0\,A_0]]
[A \cup A_1 \cup J\cup J_3 \cup\Sigma\cup JA \cup J_1]   [4 \quad c \quad .m.] [x,0,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z\leq\textstyle{1 \over 2}]
[\Delta\quad \alpha, 0, \alpha] ex [4\quad d \quad m..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B \quad \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}, \alpha] ex [4 \quad d \quad m..] [0,y,\textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, b_0]
[D \quad \alpha, \textstyle{1 \over 2}, \textstyle{1 \over 2}+\alpha] ex [4\quad d\quad m..] [\textstyle{1 \over 2},y,0]: [0\,\lt \, y\leq d_0]
[D \sim B_1]     [0,y,\textstyle{1 \over 2}]: [\textstyle{1 \over 2}-d_0=b_0\leq y\,\lt \, \textstyle{1 \over 2}]
[E \quad \alpha+\beta, \alpha, \beta] ex [4\quad d \quad m..] [0,y,z]: [[\Gamma\,Y\,H_0\,B_0\,Z]]
[EA \quad {-\alpha}+\beta, -\alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [[\Gamma\,Z_2\,B_2\,H_2\,Y]]
[F\quad \alpha+\beta, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,D_0\,G_0]]
[F \sim E_3]     [0,y,z]: [[B_2\,T_4\,H_2]]
[FA \quad {-\alpha}+\beta, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,G_2\,D_0]]
[FA \sim E_1]     [0,y,z]: [[T_2\,B_0\,H_0]]
[\Delta \cup B \cup B_1 \cup E \cup E_1 \cup EA \cup E_3]   [4\quad d \quad m..] [0,y,z]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [-\textstyle{1 \over 2}\,\lt \, z\leq\textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [8 \quad e \quad 1] [x,y,z]: [0\,\lt \, x,y\,\lt \, \textstyle{1 \over 2}]; [0 \,\lt \, z \leq \textstyle{1 \over 2}]

Table 1.5.5.6| top | pdf |
List of k-vector types for arithmetic crystal class [mm2F]: [c^{-2}>a^{-2}+b^{-2}]

See Fig. 1.5.5.6[link]. Parameter relations: [x=-\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [ y=\textstyle{1 \over 2} \alpha-\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [z=] [\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta-\textstyle{1 \over 2}\gamma].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0] ex [2 \quad a \quad mm2] [0,0,0]
[Z \quad \textstyle{1 \over 2},\textstyle{1 \over 2} ,1] ex [2 \quad a \quad mm2] [\textstyle{1 \over 2},\textstyle{1 \over 2},0]
[Z \sim Z_2]     [0,0,\textstyle{1 \over 2}]
[\Lambda \quad \alpha, \alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [0\,\lt \, z\leq\lambda_0]
[LE \quad {-\alpha}, -\alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [\lambda_2=-\lambda_0 \,\lt \, z\,\lt \, 0]
[Q \quad \textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}+ \alpha, 1] ex [2 \quad a \quad mm2] [\textstyle{1 \over 2},\textstyle{1 \over 2},z]: [0\,\lt \, z\leq q_0]
[Q \sim \Lambda_3=[\Lambda_2\,Z_4]]     [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \leq -\textstyle{1 \over 2} + q_0 = -\lambda_0]
[QA \quad \textstyle{1 \over 2}-\alpha,\textstyle{1 \over 2}- \alpha, 1] ex [2 \quad a \quad mm2] [\textstyle{1 \over 2},\textstyle{1 \over 2},z]: [q_2= -q_0\,\lt \, z\,\lt \, 0]
[QA\sim \Lambda_1=[\Lambda_0\,Z_2]]     [0,0,z]: [\textstyle{1 \over 2}-q_0=\lambda_0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[Z_2 \cup \Lambda_1 \cup \Lambda \cup \Gamma \cup LE \cup \Lambda_3]   [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[T\quad 0,\textstyle{1 \over 2},\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,0]
[T\sim T_2]     [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]
[Y \quad \textstyle{1 \over 2}, 0, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},0]
[G \quad \alpha, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,z]: [0\,\lt \, z\leq g_0]
[G \sim H_3=[H_2\,T_4]]     [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \leq -\textstyle{1 \over 2}+g_0]
[GA \quad {-\alpha}, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [\textstyle{1 \over 2},0,z]: [g_2=-g_0\,\lt \, z\,\lt \, 0]
[GA \sim H_1=[H_0\,T_2]]     [0,\textstyle{1 \over 2},z]: [\textstyle{1 \over 2}-g_0=h_0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[H \quad \textstyle{1 \over 2}+\alpha, \alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z\leq h_0]
[HA \quad \textstyle{1 \over 2}-\alpha, -\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [h_2=-h_0\,\lt \, z\,\lt \, 0]
[T_2 \cup H_1 \cup H\cup Y \cup HA \cup H_3]   [2\quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2}\,\lt \, z \leq \textstyle{1 \over 2}]
[\Sigma \quad 0, \alpha, \alpha] ex [4 \quad c \quad .m.] [x,0,0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[C \quad \textstyle{1 \over 2}, \alpha, \textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[C \sim A=[Z_2\,Y_2]]     [x,0,\textstyle{1 \over 2}]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[J \quad \alpha, \alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\, \Lambda_0\,G_0\,T]]
[JA \quad {-\alpha}, -\alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\,T\,G_2\,\Lambda_2]]
[K \quad \textstyle{1 \over 2}+\alpha, \alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,H_0\,Q_0\,Z]]
[K \sim J_3]     [x, 0 ,z]: [[Y_4\,G_2\,\Lambda_2\,Z_4]]
[KA \quad \textstyle{1 \over 2}-\alpha, -\alpha+\beta, \textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Z\,Q_2\,H_2\,Y]]
[KA \sim J_1]     [x,0,z]: [[Z_2\,Y_2\,G_0\,\Lambda_0]]
[A \cup J_1 \cup J \cup \Sigma \cup JA \cup J_3]   [4 \quad c \quad .m.] [x,0,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]; [-\textstyle{1 \over 2} \,\lt \, z \leq \textstyle{1 \over 2}]
[\Delta\quad\alpha, 0, \alpha] ex [4 \quad d\quad m..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[D\quad \alpha, \textstyle{1 \over 2}, \textstyle{1 \over 2}+\alpha] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[D \sim B]     [0,y, \textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[E \quad \alpha+\beta, \alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [[\Gamma\,Y\,H_0\,\Lambda_0]]
[EA \quad {-\alpha}+\beta, -\alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [[\Gamma\,\Lambda_2\,H_2\,Y]]
[F \quad \alpha+\beta, \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,Z\,Q_0\,G_0]]
[F \sim E_3]     [0,y,z]: [[Z_4\,\Lambda_2\,H_2\,T_4]]
[FA \quad {-\alpha}+\beta, \textstyle{1 \over 2}-\alpha, \textstyle{1 \over 2}+\beta] ex [4 \quad d \quad m..] [\textstyle{1 \over 2},y,z]: [[T\,G_2\,Q_2\,Z]]
[FA \sim E_1]     [0,y,z]: [[Z_2\,\Lambda_0\,H_0\,T_2]]
[B \cup E_1 \cup E \cup \Delta \cup EA \cup E_3]   [4 \quad d \quad m..] [0,y,z]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]; [-\textstyle{1 \over 2}\,\lt \, z\leq\textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [8 \quad e \quad 1] [x,y,z]: [0\,\lt \, x,y\,\lt \, \textstyle{1 \over 2}]; [0\,\lt \, z \leq \textstyle{1 \over 2}]

Table 1.5.5.7| top | pdf |
List of k-vector types for arithmetic crystal class [mm2F]: [a^{-2}\,\gt \, b^{-2}+c^{-2}]

See Fig. 1.5.5.7[link]. Parameter relations: [x=-\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [y=\textstyle{1 \over 2} \alpha-\textstyle{1 \over 2}\beta+\textstyle{1 \over 2}\gamma], [z=\textstyle{1 \over 2} \alpha+\textstyle{1 \over 2}\beta-\textstyle{1 \over 2}\gamma].

k-vector label, CDMLWyckoff position of IT A, cf. Section 1.5.4.3[link]Parameters
[\Gamma \quad 0,0,0] ex [2 \quad a \quad mm2] [0,0,0]
[Z \quad \textstyle{1 \over 2},\textstyle{1 \over 2} ,0] ex [2 \quad a \quad mm2] [0,0,\textstyle{1 \over 2}]
[\Lambda \quad \alpha, \alpha, 0] ex [2 \quad a\quad mm2] [0,0,z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[LE \quad {-\alpha}, -\alpha, 0] ex [2 \quad a \quad mm2] [0,0,z]: [-\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[\Gamma\cup Z\cup\Lambda\cup LE]   [2 \quad a\quad mm2] [0,0,z]: [-\textstyle{1 \over 2}\,\lt \, z\leq\textstyle{1 \over 2}]
[T\quad 1,\textstyle{1 \over 2},\textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},\textstyle{1 \over 2}]
[Y \quad \textstyle{1 \over 2}, 0, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},0]
[H \quad \textstyle{1 \over 2}+\alpha, \alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [0\,\lt \, z\,\lt \, \textstyle{1 \over 2}]
[HA \quad \textstyle{1 \over 2}-\alpha, -\alpha, \textstyle{1 \over 2}] ex [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[T \cup Y \cup H \cup HA]   [2 \quad b \quad mm2] [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2}\,\lt \, z\leq \textstyle{1 \over 2}]
[\Sigma \quad 0,\alpha,\alpha] ex [4 \quad c \quad .m.] [x,0,0]: [0\,\lt \, x\leq\sigma_0]
[U \quad 1,\textstyle{1 \over 2}+\alpha,\textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},\textstyle{1 \over 2}]: [0\,\lt \, x\,\lt \, u_0]
[U \sim \Sigma_1=[\Sigma_0\,T_2]]     [x,0,0]: [\textstyle{1 \over 2}-u_0=\sigma_0\,\lt \, x\,\lt \, \textstyle{1 \over 2}]
[A \quad \textstyle{1 \over 2}, \textstyle{1 \over 2}+\alpha, \alpha] ex [4\quad c \quad .m.] [x,0,\textstyle{1 \over 2}]: [0\,\lt \, x\,\lt \, a_0]
[C \quad\textstyle{1 \over 2}, \alpha, \textstyle{1 \over 2}+\alpha] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},0]: [0\,\lt \, x\leq c_0]
[C \sim A_1=[A_0\,Y_2]]     [x,0,\textstyle{1 \over 2}]: [a_0=\textstyle{1 \over 2} -c_0\leq x\,\lt \, \textstyle{1 \over 2}]
[J \quad \alpha, \alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\,Z\,A_0\,\Sigma_0]]
[JA \quad {-\alpha}, -\alpha+\beta, \beta] ex [4 \quad c \quad .m.] [x,0,z]: [[\Gamma\,\Sigma_0\,A_2\,Z_4]]
[K \quad \textstyle{1 \over 2}+\alpha,\alpha+\beta,\textstyle{1 \over 2}+\beta] ex [4\quad c\quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,T\,U_0\,C_0]]
[K\sim J_3]     [x,0,z]: [[T_2\,\Sigma_0\,A_2\,Y_4]]
[KA \quad \textstyle{1 \over 2}-\alpha,-\alpha+\beta,\textstyle{1 \over 2}+\beta] ex [4 \quad c \quad .m.] [x,\textstyle{1 \over 2},z]: [[Y\,C_0\,U_2\,T_4]]
[KA\sim J_1]     [x,0,z]: [[T_2\,\Sigma_0\,A_0\,Y_2]]
[A \cup A_1 \cup J \cup J_1 \cup \Sigma\cup\Sigma_1 \cup JA \cup J_3]   [4 \quad c \quad .m.] [x,0,z]: [0\,\lt \, x\,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2} \,\lt \, z\leq \textstyle{1 \over 2}]
[\Delta\quad \alpha, 0, \alpha] ex [4 \quad d \quad m..] [0,y,0]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[B \quad \textstyle{1 \over 2}+\alpha, \textstyle{1 \over 2}, \alpha] ex [4 \quad d \quad m..] [0,y,\textstyle{1 \over 2}]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}]
[E \quad \alpha+\beta, \alpha, \beta] ex [4 \quad d \quad m..] [0,y,z]: [0\,\lt \, y,z\,\lt \, \textstyle{1 \over 2}]
[EA \quad {-\alpha+\beta}, -\alpha, \beta] ex [4\quad d \quad m..] [0,y,z]: [0\,\lt \, y\,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2}\,\lt \, z\,\lt \, 0]
[\Delta\cup B\cup E\cup EA]   [4 \quad d \quad m..] [0,y,z:] [0\,\lt \, y\,\lt \, \textstyle{1 \over 2},-\textstyle{1 \over 2}\,\lt \, z\leq \textstyle{1 \over 2}]
[GP \quad \alpha,\beta,\gamma]   [8 \quad e \quad 1] [x,y,z]: [0\,\lt \, x,y\,\lt \, \textstyle{1 \over 2}, 0\leq z\,\lt \, \textstyle{1 \over 2}]
[Figure 1.5.5.5]

Figure 1.5.5.5 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [mm2F]: [a^{-2} \,\lt \, b^{-2}+c^{-2}], [b^{-2} \,\lt \, c^{-2}+a^{-2}] and [c^{-2} \,\lt \, a^{-2}+b^{-2}]. Space groups [Fmm2-C_{2v}^{18}] (42), [Fdd2-C_{2v}^{19}] (43). Reciprocal-space group ([Imm2])*, No. 44: [a^{*2} \,\lt \, b^{*2}+c^{*2}], [b^{*2} \,\lt \, c^{*2}+a^{*2}] and [c^{*2} \,\lt \, a^{*2}+b^{*2}] (see Table 1.5.5.5[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T4: [0,\textstyle{1 \over 2},-\textstyle{1 \over 2}]; Y2: [\textstyle{1 \over 2},0,\textstyle{1 \over 2}]; Y4: [\textstyle{1 \over 2},0,-\textstyle{1 \over 2}]; Z2: [0,0, -\textstyle{1 \over 2}]. Flagpoles: [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]. Wings: [x,0,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}, -\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0].

[Figure 1.5.5.6]

Figure 1.5.5.6 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [mm2F]: [c^{-2}\,\gt \, a^{-2}+b^{-2}]. Space groups [Fmm2-C_{2v}^{18}] (42), [Fdd2-C_{2v}^{19}] (43). Reciprocal-space group ([Imm2])*, No. 44: [c^{*2}\,\gt \, a^{*2}+b^{*2}] (see Table 1.5.5.6[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T4: [0,\textstyle{1 \over 2},-\textstyle{1 \over 2}]; Y4: [\textstyle{1 \over 2},0,-\textstyle{1 \over 2}]; Z4: [0,0,-\textstyle{1 \over 2}]. Flagpoles: [0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]. Wings: [x,0,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 2}], [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2}], [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0].

[Figure 1.5.5.7]

Figure 1.5.5.7 | top | pdf |

Brillouin zone with asymmetric unit and representation domain of CDML for arithmetic crystal class [mm2F]: [a^{-2}\,\gt \, b^{-2}+c^{-2}]. Space groups [Fmm2-C_{2v}^{18}] (42), [Fdd2-C_{2v}^{19}] (43). Reciprocal-space group ([Imm2])*, No. 44: [a^{*2}\,\gt \, b^{*2}+c^{*2}] (see Table 1.5.5.7[link]). The representation domain of CDML is different from the asymmetric unit. Auxiliary points: T4: [0,\textstyle{1 \over 2},-\textstyle{1 \over 2}]; Y4: [\textstyle{1 \over 2},0,-\textstyle{1 \over 2}]; Z4: [0,0,-\textstyle{1 \over 2}]. Flagpoles: [LE=[Z_4\,\Gamma] \quad 0,0,z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [HA=[T_4\,Y] \quad 0,\textstyle{1 \over 2},z]: [-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]. Wings: [JA \cup J_3] [=] [[\Gamma\,T_2\,Y_4\,Z_4]] [x,0,z]: [ 0 \,\lt \, x \,\lt \, \textstyle{1 \over 2},-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0]; [EA=[\Gamma\,Z_4\,T_4\,Y] \quad 0,y,z]: [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2},-\textstyle{1 \over 2} \,\lt \, z \,\lt \, 0].








































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